[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84960-en":3,"doc-seo-84960-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84960,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces","Spectral positional encodings for directed graphs are hindered by expensive full-spectrum magnetic eigendecompositions and by basis ambiguity of complex eigenvectors inside eigenspaces. This work proposes learnable spectral positional encodings defined as a learnable matrix function hθ(Aq) applied to random probe blocks, avoiding eigendecomposition conventions. The method is computed using Hermitian block Krylov subspaces via sparse matrix–vector products, with provable approximation depth k = O(log(1/ε)). Theory and experiments show structured response families generalize better than unconstrained eigenvalue-wise oracles.","Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces  \nJiaqing Xie  \nFudan University  \nYuxin Wang  \nFudan University  \narXiv :2607 .07032v2 [ cs .LG] 11 Jul 2026  \nAbstract  \nSpectral positional encodings (PEs) for directed graphs face two obstacles: fullspectrum magnetic methods require a dense Hermitian eigendecomposition per potential, and complex eigenvectors are defined only up to basis choices within eigenspaces, which prior work handles with basis-invariant architectures. We propose learnable spectral PEsof the form hθ (Aq) R, where Aq is a normalized magnetic operator, hθ a learnable scalar spectral response, and R a block of random probes. Because the PE is a matrix function of the operator, it is independent of eigendecomposition conventions. We compute it in a Hermitian block Krylov subspace from sparse matrix–vector products only, prove that k = O (log(1/ε)) block steps suffice uniformly over heat–resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where an unconstrained per-eigenvalue oracle overfits. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exacteigendecomposition oracle as the depth grows.  \nCross-response probe inner products provide pairwise features with 1/ √ s Monte-Carlo error, and the undirected q =0 case improves heterophilous benchmarks over no-PE and polynomial baselines.  \n1 Introduction  \nPositional encodings are a key ingredient of graph transformers and a practical route to lifting the expressive power of message-passing GNNs beyond 1-WL (Dwivedi and Bresson, 2020; Ramp´aˇsek et al., 2022) . The dominant spectral construction, Laplacian PE, uses eigenvectors of the symmetric graph Laplacian and  \ninherits three well-known problems: the cost of extracting and stabilizing individual eigenvectors, instability under spectral perturbation, and sign/basis ambiguity of eigenvectors (Lim et al., 2023; Wang et al., 2022; Huang et al., 2024) . Learnable spectral PEs (LLPE) replace raw eigenvectors with a trainable filter of the spectrum (Ito et al., 2025), which improves task adaptivity but still requires a spectral representation on which the filter is applied.  \nDirected graphs sharpen every one of these issues. Direction matters in citation, program, and circuit graphs, yet the adjacency matrix is no longer symmetric, and the standard fix of symmetrization destroys the information of interest. The magnetic Laplacian (Geisleret al., 2023) restores symmetry in the form of a complex Hermitian operator Lq whose phases encode edge directions, and Multi-q magnetic PEs (Huang et al., 2025) show that a set of potentials q is provably necessary to express directed walk profiles. Practical eigenvectorbased magnetic PEs use a partial Hermitian eigensolver per potential, while exact full-spectrum pairwise readouts require a dense decomposition; in both cases, complex eigenvectors are defined only up to per-eigenspace unitary gauge, which Huang et al. (2025) address with dedicated basis-invariant architectures.  \nThis paper. We take a different route. We define the PE directly as a learnable matrix function of the  \nmagnetic operator applied to a block of random probes, Zq (θ) = hθ (Aq)R ∈ Cn×s, R ∼ CN(0, ~~1~~sI), (1)  \nand approximate it in a Hermitian block Krylov subspace Kk (Aq , R) using sparse matrix–vector products only. This design has three consequences. The probes are part of the randomized encoding: for a fixed draw, equivariance is conditional on permuting R with the nodes; if probes are freshly sampled after relabeling, the encoding is permutation equivariant in distribution because the Gaussian law is exchangeable (Proposition 2) . Here “gauge invariance” refers only to eigenbasis choices. Under a vertex-wise magnetic gauge, the PE is covariant if R co-transforms.","cbCails3mt99ujNR","https://ap.wps.com/l/cbCails3mt99ujNR","pdf",402224,2,1,12,"English","en",105,"# Abstract\n# 1 Introduction\n## Positional encodings for graph transformers\n## Laplacian PE issues\n## Learnable spectral PE and directed graphs\n## Magnetic Laplacian and gauge/basis ambiguity\n## Proposed learnable matrix-function PE via Krylov subspaces\n## Consequences: independence, cheap approximation, and controlled estimation\n## Empirical setup and results","[{\"question\":\"What are the main challenges of spectral positional encodings for directed graphs?\",\"answer\":\"They require either dense Hermitian eigendecompositions for full-spectrum magnetic methods or they suffer from basis ambiguity because complex eigenvectors are only defined up to unitary choices within eigenspaces.\"},{\"question\":\"How does the proposed method define positional encodings without eigendecomposition conventions?\",\"answer\":\"It defines the positional encoding as Zq(θ) = hθ(Aq)R, a matrix function of a normalized magnetic operator applied to random probes, so eigenvectors and their basis choices never explicitly appear.\"},{\"question\":\"How are the matrix-function positional encodings computed efficiently?\",\"answer\":\"They are approximated in a Hermitian block Krylov subspace using only sparse matrix–vector products, with a depth bound k = O(log(1/ε)) that holds uniformly for learnable response families.\"}]",1784199714,30,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"eigenbasis-independent-learnable-spectral-positional-encodings-for-directed-graphs-via-hermitian-block-krylov-subspaces","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/eigenbasis-independent-learnable-spectral-positional-encodings-for-directed-graphs-via-hermitian-block-krylov-subspaces/84960/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are the main challenges of spectral positional encodings for directed graphs?","Question",{"text":75,"@type":76},"They require either dense Hermitian eigendecompositions for full-spectrum magnetic methods or they suffer from basis ambiguity because complex eigenvectors are only defined up to unitary choices within eigenspaces.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method define positional encodings without eigendecomposition conventions?",{"text":80,"@type":76},"It defines the positional encoding as Zq(θ) = hθ(Aq)R, a matrix function of a normalized magnetic operator applied to random probes, so eigenvectors and their basis choices never explicitly appear.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the matrix-function positional encodings computed efficiently?",{"text":84,"@type":76},"They are approximated in a Hermitian block Krylov subspace using only sparse matrix–vector products, with a depth bound k = O(log(1/ε)) that holds uniformly 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